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Maoduan Ran (talk | contribs) |
Maoduan Ran (talk | contribs) |
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provided <math>a^{ij}(x) = \sum_{k=1}^d \sigma^i_k(x)\sigma^j_k(x)</math>, and <math>\sigma^{ij}(x)</math>, <math>b^i(x)</math> are Lipschitz continuous. By Itô's lemma, for <math>f \in C^{2,1}(\mathbb{R}^d \times [\tau,\infty))</math> we have <math>df(X_t,t) = \Bigl(\tfrac{\partial f}{\partial t} + \sum_{i=1}^d b^i \tfrac{\partial f}{\partial x_i} + \nu \sum_{i,j=1}^d a^{ij}\tfrac{\partial^2 f}{\partial x_i \partial x_j}\Bigr)\,dt + \text{(martingale terms)}.</math>
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The infinitesimal generator <math>\mathcal{A}</math> of <math>X_t</math> is defined for <math>f \in C^{2,1}(\mathbb{R}^d \times \mathbb{R}^+)</math> by <math>\mathcal{A}f(\mathbf{x},t) = \sum_{i=1}^d b^i(\mathbf{x},t)\,\tfrac{\partial f}{\partial x_i} + \nu \sum_{i,j=1}^d a^{ij}(\mathbf{x},t)\,\tfrac{\partial^2 f}{\partial x_i \partial x_j} + \tfrac{\partial f}{\partial t}.</math>
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